- Let \( \displaystyle c : I \subset \mathbb{R} \longrightarrow \mathbb{R}^{3} \) be a closed curve parametrized by arc length and with curvature \( \displaystyle \kappa(s)>0 \, , \, \forall s \in I \). If the curve \( \displaystyle \gamma : I \subset \mathbb{R} \longrightarrow S^{2} \, , \, \gamma (s) = \vec{n} (s) \) is simple and defines a simple region on \( \displaystyle S^{2} \), then it divides \( \displaystyle S^{2} \) in two regions with equal areas.
- Let \( \displaystyle S \subset \mathbb{R}^{3} \) be a regular surface homeomorphic to a sphere, with positive Gaussian curvature. Let \( \displaystyle \Gamma \subset S \) be a simple closed geodesic in \( \displaystyle S \), and let \( \displaystyle A \) and \( \displaystyle B \) be the two regions of \( \displaystyle S \) which have \( \displaystyle \Gamma \) as a common boundary. Let \( \displaystyle N : S \longrightarrow S^2 \) be the Gauss map of \( \displaystyle S \) and suppose that \( \displaystyle N \) is 1-1. Prove that \( \displaystyle N(A) \) and \( \displaystyle N(B) \) have the same area.
Equal area
-
- Community Team
- Posts: 314
- Joined: Tue Nov 10, 2015 8:25 pm
Equal area
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 1 guest